Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks, where highly overparameterized models escape overfitting and achieve good test performance, at odds with the standard bias-variance trade-off described by statistical learning theory. In the present work, we explore a link between this phenomenon and the increase of complexity and sensitivity of the function represented by neural networks. In particular, we study the Boolean mean dimension (BMD), a metric developed in the context of Boolean function analysis. Focusing on a simple teacher-student setting for the random feature model, we derive a theoretical analysis based on the replica method that yields an interpretable expression for the BMD, in the high dimensional regime where the number of data points, the number of features, and the input size grow to infinity. We find that, as the degree of overparameterization of the network is increased, the BMD reaches an evident peak at the interpolation threshold, in correspondence with the generalization error peak, and then slowly approaches a low asymptotic value. The same phenomenology is then traced in numerical experiments with different model classes and training setups. Moreover, we find empirically that adversarially initialized models tend to show higher BMD values, and that models that are more robust to adversarial attacks exhibit a lower BMD.

Owen and Hoyt recently showed that the effective dimension offers key structural information about the input-output mapping underlying an artificial neural network. Along this line of research, this work proposes an estimation procedure that allows the calculation of the mean dimension from a given dataset, without resampling from external distributions. The design yields total indices when features are independent and a variant of total indices when features are correlated. We show that this variant possesses the zero independence property. With synthetic datasets, we analyse how the mean dimension evolves layer by layer and how the activation function impacts the magnitude of interactions. We then use the mean dimension to study some of the most widely employed convolutional architectures for image recognition (LeNet, ResNet, DenseNet). To account for pixel correlations, we propose calculating the mean dimension after the addition of an inverse PCA layer that allows one to work on uncorrelated PCA-transformed features, without the need to retrain the neural network. We use the generalized total indices to produce heatmaps for post-hoc explanations, and we employ the mean dimension on the PCA-transformed features for cross comparisons of the artificial neural networks structures. Results provide several insights on the difference in magnitude of interactions across the architectures, as well as indications on how the mean dimension evolves during training.